Plainly put...YES!
The automatic carwash uses a very strong detergent/striper in order to remove the dirt and grime from your vehicle without. So just imagine what it is doing to your finish. In doing so, this detergent will strip the wax off the paint exposing the clear coat to the harsh elements. Once the protective wax is stripped away, and you continue taking your vehicle through these cheap automatic carwashes the harsh chemicals will eventually eat away at your vehicle's clear coat, black trim molding, and eventually, the paint...stripping away its glossy finish.